Geometric chain rule proof and question about area formula for polar coordinate

ok i remember i took notes for the proof of chain rule and the teacher used triangles and stuff so it was a lot of geometry which i didn't understand at the time.. unfortunately i threw it out and i cannot for the life of me find it ANYWHERE on the internet.. does anyone know what i'm talking about? can you help me out T_T

i also don't remember the proof for the area formula for polar functions.. integral a to b of 1/2r^2 d theta .. does anyone have the proof for it?